Deformation and Element Methods for Frames

Abstract

In the deformation method for frames the structure is considered as formed by individual beams, connected at nodes. Initially all nodes are considered constrained, and they are then released one at a time. The correct solution is obtained by finding the combination of node displacements that do not require any additional constraining forces at the nodes. This approach has a number of advantages. First, the release of a constraint only affects beams directly connected to the constraint, and the equations of the method therefore only require a local analysis. Furthermore, this local analysis only involves individual beams, and it can therefore easily be given a general systematic form, suitable for computer analysis. This so-called finite element formulation is quite general and can be developed from the principle of virtual work for plates, shells and solid bodies as well. The chapter covers the classic deformation method of frames, intended for hand calculation, as well as the finite element method for frame structures. The finite element formulation for elastic frames is obtained by rearranging the procedure of the deformation method into a systematic matrix format. The basic idea is to represent each beam as an element with a stiffness matrix, including all displacement components at each of its nodes. These elements are then assembled into a frame structure, and the nodal displacements are determined by solution of the corresponding equation system. The finite element formulation has been implemented for plane frames in the MATLAB code MiniFrame, which is described in detail.